TY - JOUR

T1 - A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

AU - Feng, Bao Feng

AU - Maruno, Ken Ichi

AU - Ohta, Yasuhiro

N1 - Funding Information:
BF appreciates the comments and discussions with Professor Youjin Zhang and Professor Qingping Liu and the partial support by the National Natural Science Foundation of China (No. 11428102). The work of KM is partially supported by JSPS Grant-in-Aid for Scientific Research (C-15K04909) and CREST, JST. The work of YO is partly supported by JSPS Grant-in-Aid for Scientific Research (B-24340029, C-15K04909) and for Challenging Exploratory Research (26610029).

PY - 2017/1/4

Y1 - 2017/1/4

N2 - In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.

AB - In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.

KW - BKP and modified BKP hierarchy

KW - hodograph and discrete hodograph transform

KW - integrable discretization

KW - pseudo 3-reduction

KW - short wave model of two-component Degasperis Procesi (DP) equation

KW - two-component reduced Ostrovsky equation

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U2 - 10.1088/1751-8121/50/5/055201

DO - 10.1088/1751-8121/50/5/055201

M3 - Article

AN - SCOPUS:85010060695

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 5

M1 - 055201

ER -